Nuclear quantum memory for hard x-ray photon wave packets

Optical quantum memories are key elements in modern quantum technologies to reliably store and retrieve quantum information. At present, they are conceptually limited to the optical wavelength regime. Recent advancements in x-ray quantum optics render an extension of optical quantum memory protocols to ultrashort wavelengths possible, thereby establishing quantum photonics at x-ray energies. Here, we introduce an x-ray quantum memory protocol that utilizes mechanically driven nuclear resonant 57Fe absorbers to form a comb structure in the nuclear absorption spectrum by using the Doppler effect. This room-temperature nuclear frequency comb enables us to control the waveform of x-ray photon wave packets to a high level of accuracy and fidelity using solely mechanical motions. This tunable, robust, and highly flexible system offers a versatile platform for a compact solid-state quantum memory at room temperature for hard x-rays.

to residual magnetic fields or electric field gradients.From the fitted foil thickness an optical thickness ζ (also called effective thickness for nuclear resonance scattering) can be calculated, defined by (40), with σ 0 = 2557.67kbarn being the nuclear resonant cross-section, f LM = 0.76 the Lamb-Mössbauer factor, ρ n = 4.35⋅10 28 m -3 the number density of the resonant isotope 57 Fe and  the derived foil thickness.The optical thickness is the relevant quantity to describe nuclear resonant interactions as it includes the elastic-to-inelastic scattering ratio (Lamb-Mössbauer factor), the strength of coherent scattering (number density and foil thickness) and the internal conversion process [included in the nuclear resonant cross-section (40)].The parameters for all foils as well as the mean values are given in Table S1.The small standard deviations of the mean values show that the foils only differ little.Therefore, the absorption line shapes of the frequency comb are nearly identical, which would otherwise undermine the comb's efficiency.

Thin film cavity characterization
In the grazing incidence setup at the synchrotron, the reflected intensities of the non-resonant part (X-ray reflectometry) and the resonant, delayed part (nuclear X-ray reflectometry) of the incoming 14.41 keV X-ray pulse were detected as a function of the incidence angle.While the former gives information about the electron density distribution inside the thin film cavity, the latter accesses the distribution of the resonant nuclei inside the sample.Together, the thickness and roughness of each layer can be derived by using a transfer matrix approach (54) (implemented in the software package Nexus).The experimental data and the calculated intensity are shown in Fig. S2A and B. The derived layer parameters are listed in Table S2.The hyperfine parameters can be obtained from the decay histogram, which is shown in Fig. S2C for two different incident angles: 0.1505° (the first order waveguide mode) and 0.45°.The fit results are listed in Table S3.For the quadrupole splitting, a Gaussian distribution with only positive values is assumed.To cross-check the derived parameters, the cavity setup is combined with a stainless-steel foil mounted on a Mössbauer transducer.A two-dimensional histogram, similar to the frequency comb measurement, containing information in both time and energy domain can be recorded by Doppler shifting the relatively narrow absorption line of the stainless-steel foil.This interferometric setup is reported to be able to access the energy spectrum of the cavity, if the cavity is illuminated at the critical coupling condition of a cavity minimum and the two-dimensional histogram is time integrated in a late time window (29).This is shown in Fig. S2D, where the integration time window was set between 69 ns and 165 ns.The shape and width of the calculated spectrum match the experimental data very well.For a good match, an isomer shift of 0.3 mm/s relative to the stainless-steel absorber must be assumed.The deviation from the known isomer shift between -Fe and stainless-steel of 0.1 mm/s (55), is likely caused by the disordered nature of the 57 Fe atoms in the ultrathin iron layer consisting of only three to four atomic layers.
Influence of the number of foils in the frequency comb In the main text, the frequency comb consists of seven 57 SS foils, i.e. seven teeth.To investigate the influence of the teeth number  on the frequency comb, stainless-steel foils were removed or added at the end of the comb, so that the remaining teeth stay always spectrally equidistant.In Fig. S3A, the temporal responses for a fixed velocity spacing of 5 mm/s are shown for frequency combs consisting of five to eight foils.Several trends are clearly visible as more comb teeth are introduced:  the echoes get sharper in time, the intensity contrast between the echoes and their side maxima rises and the number of these side maxima increases by one with every newly added foil.In total, there are always  − 2 side maxima and  − 1 side minima.The observed characteristics strongly resemble diffraction intensity patterns known from spatial gratings.Here, however, the frequency comb constitutes a (single-photon) grating in the frequency domain.Translating the known diffraction intensity distribution from spatial gratings into a frequency grating by adding up  complex plane waves with different but equidistant frequencies, the intensity of the emitted photon wave packet should follow, with ∆ = ℏ∆ being the comb's frequency spacing caused by the Doppler shift.An exponential factor is added to account for the temporal response of a single comb tooth with spectral width Γ.
The width of an absorber with optical thickness  is roughly given by Γ = Γ 0 (1 + /4) (54), thus about 8 Γ 0 in our case.The actual temporal response of a single comb tooth, i.e. of a stainless-steel foil, is shown in Fig. S1 which is modified by a dynamical beat node at 70 ns.
As shown in Fig. S3A, the intensity distribution defined by Eq. (S3) reproduces the main features of the experimental data very well.More specifically, the position and width of the echoes are matching nearly perfectly, as well as the number of side maxima as a function of the teeth number.This indicates the excellent precision of the frequency comb setup.If one of the velocity transducers moves with a 5%-off velocity, the frequency comb formation is severely disturbed, as shown by simulations depicted in Fig. S3B.With such an imperfect velocity setting, the distinct side maxima structure between the echoes as visible in the measurements would not be noticeable anymore.

Estimation of losses in the storage process
The photons propagating through the frequency comb experience not only coherent resonant forward scattering, but also Compton scattering, photoelectric absorption, and incoherent resonant processes, such as inelastic phonon excitations, internal conversion, and nuclear fluorescence.Photons involved in these incoherent processes are lost for the storage process and do not appear in the measured histogram which counts only photons scattered in the forward direction.Hence, the incoherent processes are not included in the normalization factor  FS used for the echo efficiency  echo calculation.To estimate the incoherent losses of a nuclear system (e.g. the frequency comb), a direct relation between the incoming and outgoing photon pulse energy must be established.This Table S3.Hyperfine parameters of X-ray cavity.Incidence angle, divergence and hyperfine parameters derived from the temporal beat pattern in Fig. S2B and the energy spectrum in Fig. S2D.The isomer shift is relative to the stainless-steel foil used as a reference absorber.For energy conversion: 0.097 mm/s = 1Γ 0 = 4.66 neV.can be done using the analytical response function of a thick nuclear resonant scatterer in the absence of hyperfine interactions (56),

Data
with   being the electronic absorption coefficient due to Compton scattering and photoelectric absorption (for stainless-steel at the 57 Fe resonance energy:   ≃ 0.048 μm −1 ), () the Dirac delta distribution,  0 the natural lifetime of the excited state ( 0 = 141 ns),   the motion-induced Doppler shift (  = Δ  /ℏ) and () the Heaviside step function.The function,  1 �√�/√, with its argument  = ζ/ 0 , contains the Bessel function of first kind,  1 , which leads to an aperiodic dynamical beating for a medium with a large enough optical thickness.The optical thickness ζ carries information about the strength of incoherent-inelastic scattering via the Lamb-Mössbauer factor and of the internal conversion process which is included in the nuclear resonant cross section [Eq.( S3)].
With a photon with complex wave packet amplitude ℰ in impinging, and the foil's response function given, the complex photon wave packet emitted from the foil into the coherent resonant forward scattering channel is obtained by ( 57) where the operator " * " denotes a convolution.Eq. (S5) can be applied iteratively to calculate the propagation through multiple foils, namely through the seven Doppler detuned foils of the frequency comb.As a measure of the field energy, the time integrated intensity, , can be used to determine the total losses  tot [defined in Eq. ( 4)] via, In our case, the emitted wave packet from the superradiant cavity state, ℰ cav , is selected as incoming photon wave packet, ℰ in , since it is desired to be stored in the frequency comb.Note that the 3B and S2C).However, with the layer structure and hyperfine parameters determined via the fit routines, the complex amplitudes ℰ cav can be calculated by Nexus.In this calculation, the nonresonant interaction of the cavity with the synchrotron pulse is neglected, since only the resonant part is of interest for the storage procedure.
With ℰ cav determined from the simulation, the full propagation of ℰ cav through the frequency comb is calculated using Eq.(S5).The result is shown in Fig. S4A for a velocity spacing of 3.1 mm/s.The overall intensity scaling depends on an arbitrary scaling of ℰ cav , which is here chosen to be 1/� cav , so that the incoming energy is normalized to one.The relative scaling between ℰ out and ℰ cav is dictated by the analytical response function [Eq.(S4)], and therefore, is deterministic.The emitted photon wave packet can also be calculated using the numerical energy dependent scattering amplitudes obtained by Nexus.With the non-resonant contribution removed from the cavity scattering amplitude, the Fourier transformation of the product of all scattering amplitudes along the beam path (cavity and foils) results in a nearly identical outcome as obtained via the analytical response function, as shown in Fig. S4A.
The total losses  tot as a function of the frequency comb's velocity spacing is shown in Fig. S4B.The overlap of the Doppler detuned absorption lines of the frequency comb depends on the velocity spacing.It is therefore not surprising that the coherent forward scattering intensity, and thus also its losses, strongly varies in the region where the lines strongly overlap (in between 0 mm/s and 1.5 mm/s).For higher velocities, the losses saturate, with a lower limit imposed by the non-resonant part,  el , which is 66% for the given foil thicknesses at the 57 Fe resonance energy.
The calculated total losses were used to obtain the quantum memory efficiency depicted in Fig. 4A in the main text.For the derived optimal quantum memory performance at 3.1 mm/s, the total losses amount to  tot = 74%.

Fig
Fig. S2.X-ray cavity characterization measurements.Measurements on the thin film cavity.(A) Reflected intensity with only the non-resonant (non-delayed) photons counted.(B) Reflected intensity with only the resonant (delayed by >13 ns) photons counted.(C) Temporal beat pattern at the incidence angle 0.1505°, where the first order waveguide mode is excited [marked in (A) and (B) as well], and at 0.45°.(D) Energy spectrum at the incidence angle 0.1505° of the combined setup with a stainless-steel foil mounted on a Mössbauer transducer, obtained by an integration in the time window from 69 ns to 165 ns.Theoretical curves (red and blue solid lines) are either the numerical fits (A, C) or simulations (B, D) with parameters obtained from the other fits.

Fig. S3 .
Fig. S3.Additional NFC characterization.(A) Measured decay histograms of frequency comb setups with varying number of foils, i.e. comb teeth, with a fixed velocity spacing of 5 mm/s (black).The decay patterns closely follow diffraction intensity patterns from a grating, see Eq. (S3) (gray, solid line).(B) Simulated decays of the frequency comb consisting of seven comb teeth with a fixed velocity spacing of 5 mm/s, wherein the drive supposed to run at 10 mm/s, is detuned by 0% ("perfect"), 1% or 5%.

Fig. S4 .
Fig. S4.Estimation of total losses.(A)Intensities of the photon wave packet emitted from the cavity decay, ℰ cav , and from the combined setup (cavity and comb), ℰ out , calculated via the analytical response function.The simulation of the emitted wave packet using Nexus is shown as a comparison.(B) Total energy loss after passing through the frequency comb calculated via the ratio of the time integrated outgoing to incoming intensity as a function of the velocity spacing.A lower limit is caused by the non-resonant electronic interaction,  el = 0.66.

Table S1 . Results of stainless-steel foils characterization. Fitted
foil thicknesses and hyperfine parameters for each stainless-steel foil.The mean values are also given together with their standard deviation.For energy conversion: 0.097 mm/s = 1Γ 0 = 4.66 neV.

Table S2
. Results of X-ray cavity characterization.Cavity layer structure derived from the non-resonant reflectivity measurement in Fig.S2A.